Generic representation theory of finite fields in nondescribing characteristic

Abstract Let Rep ( F ; K ) denote the category of functors from finite dimensional F -vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and char F is invertible in K, then the K-linear abelian category Rep ( F ; K ) is equivalent to the product, over all n ≥ 0 , of the categories of K [ GL n ( F ) ] -modules. As a consequence, if K is also a field, then small projectives are also injective in Rep ( F ; K ) , and will have finite length. Even more is true if char K = 0 : the category Rep ( F ; K ) will be semisimple. In the last section, we briefly discuss ‘ q = 1 ’ analogues and consider representations of various categories of finite sets. The main result follows from a 1992 result by L.G. Kovacs about the semigroup ring K [ M n ( F ) ] .